Dr. Junchao Xia Center of Biophysics and Computational Biology. Fall /21/2016 1/23

Transcription

1 BIO53 Bosascs Lcur 04: Cral Lm Thorm ad Thr Dsrbuos Drvd from h Normal Dsrbuo Dr. Juchao a Cr of Bophyscs ad Compuaoal Bology Fall

2 Iroduco I hs lcur w wll alk abou ma cocps as lsd blow, pcd valu ad varac of fucos of radom varabls Lar combao of radom varabls Moms ad mom grag fuco Thorcal drvao of h cral lm horm Thr dsrbuos drvd from h ormal dsrbuo 906 3

3 Masurs of locao ad sprad ca b dvlopd for a radom varabl much h sam way as for sampls. Th aalog of h arhmc ma s calld h pcd valu of a radom varabl, or populao ma, ad s dod by or ad rprss h avrag valu of h radom varabl. pcd valu of a bomal dsrbuo,p = p pcd valu of a Posso dsrbuo wh pcd valu of a ormal dsrbuo wh pcd Valu of a Radom Varabl pcd Valu of a Radom Varabl 906 d f p k k k k k k k k k k 0 0!!!, N 0 dz dz z d z z z 33

4 Th aalog of h sampl varac s for a radom varabl s calld h varac of a radom varabl, or populao varac, ad s dod by Var or. Th sadard dvao of a radom varabl, dod by sd or, s dfd by h squar roo of s varac. Varac of a bomal dsrbuo,p = pq Varac of a Posso dsrbuo Varac of a ormal dsrbuo Varac of a Radom Varabl Varac of a Radom Varabl 906 d f p Var ] [ } ] {[, N 0 ] [ du u dz z d Var u u z z z 43

5 pcd Valu ad Varac of Fucos of Radom Varabls pcd valu of fucos of radom varabls Y g, f Y Y Y [ g h Y ] a ad b ar cosas Y Y, a b a b g Varac of fucos of radom varabls g p g f d ad Y ar dpd radom varabls, Y g, Var Y a ad b ar cosas Var a b {[ a b h y p for ay radom varabls dpd or o p y ad Y ar dpd radom varabls, Var Y Var VarY [ g ] [ h Y ] [ g [ g ]] p [ g [ g ]] f d a b ] } { b [ ] } b Var

6 Sums or dffrc or mor complcad lar fucos of radom varabls hr couous or dscr ar of usd. A lar combao L of h radom varabls, s dfd as ay fuco of h form L = c + +c. A lar combao also calld a lar coras. To compu h pcd valu ad varac of lar combaos of radom varabls, w us h prcpl ha h pcd valu of h sum of radom varabls s h sum of h rspcv pcd valus. No mar L c, dpd o ach or o, L Varac of L whr,, ar dpd s L Lar Combaos of Radom Varabls c, VarL c c Var c c Th corrspodg cas for dpd radom varabls wll b show lar

7 Covarac of Dpd Radom Varabls Th covarac bw wo radom varabls ad Y s dod by Cov,Y ad s dfd by Cov, Y [ Y Y] [ Y ] Y Y Y Y Y Y y y whr s h avrag valu of, y s h avrag valu of Y, ad Y = avrag valu of h produc of ad Y. Cov a, Y Cov, Y, Cov a, by abcov, Y, Cov, Y Z Cov, Y Cov, Z If ad Y ar dpd, h h covarac bw hm s 0. If larg valus of ad Y occur amog h sam subjcs as wll as small valus of ad Y, h h covarac s posv. If larg valus of ad small valus of Y or covrsly, small valus of ad larg valus of Y d o occur amog h sam subjcs, h h covarac s gav. To oba a masur of rladss or assocao bw wo radom varabls ad Y, w cosdr h corrlao coffc, dod by Corr,Y or ad s dfd by = Corr,Y = Cov,Y y whr ad y ar h sadard dvaos of ad Y, rsp. I s a dmsolss quay ha s dpd of h us of ad Y ad rags bw - ad. y y y

8 ampls of Corrlad Radom Varabls If ad Y ar appro. larly rlad, a corrlao coffc of 0 mpls dpdc. Corrlao coffc clos o mpls arly prfc posv dpdc wh larg valus of corrspodg o larg valus of Y ad small valus of corrspodg o small valus of Y. Corr clos o - mpls prfc gav dpdc, wh larg valus of corrspodg o small valus of Y ad vc vrsa

9 Lar Combaos of Dpd Radom Varabls To compu h varac of a lar coras volvg wo dpd radom varabls ad, w ca us h gral quao Varac of lar combao of radom varabls gral cas Th varac of h lar coras

10 I sascs, hghr-ordr sascs volvs usg h hrd or hghr powr of a sampl such as h hrd or hghr moms as dfd blow Th ormalsd m-h cral mom or sadardzd mom s h m-h cral moms dvdd by m. Moms ad Hghr Moms ad Hghr-Ordr Sascs Ordr Sascs 906 ] [ h cral moms -, m m m m m m d f p m m m ] [ h ormalsdcral moms - 03

11 Skwss ad Kuross Th ormalsd hrd cral mom s calld h skwss, of dod by γ. [ ] for ormal dsrbuo Th ormalsd four cral mom or sadardzd mom s calld Kuross, of dod by [ ] Th css kuross s dfd as kuross mus 3. Th css kuross = 0, for a ormal dsrbuo. Th css kuross > 0, far als ha a ormal dsrbuo. Th css kuross < 0, hr als ha a ormal dsrbuo

12 Samplg Dsrbuo How s a spcfc radom sampl, usd o sma μ ad σ, h ma ad varac of h udrlyg dsrbuo? A aural smaor o slc umbr for h squc ad ak h avrag, s a sgl ralzao of a radom varabl ovr all possbl sampls of sz ha could hav b slcd from h populao. dos a radom varabl, ad dos a spcfc ralzao of h radom varabl a sampl. Th samplg dsrbuo of s h dsrbuo of valus of ovr all possbl sampls of sz ha could hav b slcd from h rfrc populao

13 ampl of Samplg Dsrbuo Frqucy dsrbuo of h sampl ma from 00 radomly slcd sampls of sz 0. Th pcd valu of ovr s samplg dsrbuo s qual o

14 Cral Lm Thorm L,, b a radom sampl from som populao wh ma ad varac. Th for larg, ~ N, v f h udrlyg dsrbuo of dvdual obsrvaos h populao s o ormal. Th symbol ~ s usd o rprs appromaly dsrbud. Ths horm allows us o prform sascal frc basd o h approma ormaly of h sampl ma dsp h oormaly of h dsrbuo of dvdual obsrvaos. Th skwss of h dsrbuo ca b rducd by rasformao daa usg log scal. Th cral-lm horm ca hm b applcabl for smallr szs ha f h daa ar rad h orgal scal

15 ampls for Cral Lm Thorm

16 Mom-Grag Fuco Th mom-grag fuco mgf of a radom varabl s dfd as M p Propry A: If h mom-grag fuco ss for a op rval coag zro, uquly drm h probably dsrbuo. Propry B: f h mom-grag fuco ss a op rval coag zro, hm h -h drvav a =0, M 0= Propry C: If has h mgf M ad Y=a+b, h Y has h mgf M Y = a M b. Propry D: f ad Y ar dpd radom varabls wh mgf s M ad M Y ad Z=+Y, hm M Z =M M Y o h commo rval whr boh mgf s s. f d

17 Th mom-grag fuco mgf of h sadard ormal dsrbuo s dfd as Mom Mom-Grag Fuco of Sadard Normal Dsrbuo Grag Fuco of Sadard Normal Dsrbuo " 0 0 ' hrfor, Sc 0 0 Var M M du d M d M u u 73

18 Thorcal Drvao of Cral Lm Thorm S L Z Sc S M S [ M ] ad M Z [ M ] Ms has a Taylor srspaso abou zro: Ms M 0 sm'0 Sc M W hav, M Z a I ca b show ha f a a, h, lm From hs rsul,w g M whr S, whr, s a sum of dpd radom varabls, 0, M 0, M '0 0, M ''0, as, W d o show h mgf Z s h mgf Z ar dpd ad dcally dsrbuo from0, ds o h mgf s M ''0 s whr whr as, 0 of h sadard ormal dsrbuo. of sadard ormal dsrbuo. s s a 0 as as s 0 0, ad

19 To oba a rval sma for, a w famly of dsrbuos, calld ch-squar dsrbuos, mus b roducd o abl us o fd h samplg dsrbuo of S from sampl o sampl. If Z s a sadard ormal radom varabl, h dsrbuo of U=Z calld h ch-squar dsrbuo wh dgr of frdom, dod by. If ~ N,, h - ~ N0,, ad hrfor [- ] ~. If U, U,.. U ar dpd ch-squar radom varabls wh dgr of frdom, h dsrbuo of V = U +U +..U s calld chsquar dsrbuo wh dgrs of frdom ad s dod by, wh h pdf as blow, Ch Ch-Squar Dsrbuo Squar Dsrbuo V Var V M d g v v v v f v,, 0, fuco Gamma 0, dsy fuco Gamma 0, g 0,,

20 ampl of Ch-Squar Dsrbuo If whr,, ~ N0, ad h s ar dpd, h G s sad o follow a ch-squar dsrbuo wh dgrs of frdom df. Th dsrbuo s of dd by. Th ch-squar dsrbuo oly aks o posv valus ad s always skwd o h rgh. For 3, h dsrbuo has a mod grar ha 0 ad s skwd o h rgh. Th skwss dmshs as crass. Th uh prcl of a d dsrbuo ha s, a ch-squar dsrbuo wh d df s dod by d whr Pr d < d u

21 Sud s Dsrbuo If Z ~ N0, ad U ~, ad Z ad U ar dpd, h h dsrbuo of W Z U s calld dsrbuo wh dgrs of frdom, solvd by a sasca amd Wllam Goss Sud. Th dsrbuo s o uqu bu s a famly of dsrbuos dd by a paramr, h dgrs of frdom df of h dsrbuo. [ ] f Th dsy fucuo wh dgrs of frdom s If,, ~ N, ad ar dpd radom varabls, h - s s dsrbud as a dsrbuo wh - df. Th dsrbuo wh d dgrs of frdom s rfrrd o as h d dsrbuo. Th 00 uh prcl of a dsrbuo wh d dgrs of frdom s dod by d,u, ha s Prd < d,u u f f, f ormal dsrbuo 906 3

22 F Dsrbuo L U ad V b dpd ch-squar radom varabls wh m ad dgrs of frdom rspcvly. Th dsrbuo of a w radom varabl W=UmV s calld h F dsrbuo wh m ad dgrs of frdom ad s dod by F m,, wh h dsy fuco as f w [ m ] m m m w m m w m, w

23 Summary I hs chapr, w dscussd pcd ad varac of fucos of radom varabls Mom grag fuco Cral-lm horm Thr dsrbuos from h ormal dsrbuo, ch-squar, ad F dsrbuos

24 Th d